The main objective of this dissertation is the analysis of solutions to a class of linear and non-linear parabolic partial differential equations (PDEs) in phase space for dimensions greater or equal than three. In particular, these equations can be used to describe diffusion dynamics in a relativistic setting. This study is motivated not only by the vast range of applications of diffusion models, but also by the still poor understanding of the mathematical techniques involved in this study. In fact, the diffusion term in the models to be considered is non-uniformly elliptic and spatially degenerate, i.e., some spatial derivatives are absent in the diffusion operator. Moreover, for some of the models studied in this
thesis, the coefficients of the diffusion equation depend on the time variable. These properties distinguish the models under discussion from the other diffusion models studied in the literature and warn that the standard techniques for
parabolic PDEs might not apply to our framework.